You can change the medium, but it does change how the cells behave and also selects for populations of cells that can survive the changes - which means that you are no longer working with the cell line you thought you were, you are now working with a derivative of that line!

You can wean (slow change from one medium to another by using different ratios of each medium) cells onto different sorts of medium, but you have to be very careful to characterize the cells after doing so to ensure that nothing important to your experiments has happened.